Detailed six-level model - LIF QUANTIFICATION

CHAPTER 4 LIF QUANTIFICATION

4.2 Detailed six-level model

In order to examine the detailed effects of rotational, electronic and vibrational energy transfer (RET, EET and VET respectively) as well as temporally varying excitation, a minimum of a six-level transient rate equation model is required to adequately simulate the transient dynamics for the OH A-X (1,0) system. The rate equation approach has been validated in Daily [200] and also in Settersten and Linne [201] under the assumption that the coherence dephasing time is significantly smaller than the laser time pulse time scale;

this is the case for nanosecond excitation and atmospheric pressures examined here.

When combined with the fact that four out of the six levels in the detailed six-level model represent multiple rotational levels or 'lumped' levels, the error introduced by the rate equation approach is negligible. The approach of lumping several rotational levels in this study is justified in the view that the spectrally integrated LIF signal is considered as the validation parameter, where as the spectrally dispersed signal is not considered.

The results of Daily and Rothe [202] show that a minimum of a six-level model is necessary to adequately study the effects of temperature on OH LIF for (1,0) excitation.

By using a six-level model, Daily and Rothe [202] show that important effects such as RET and VET are necessary to be included in any model for accurate predictions, thus predictions utilizing a two-level model can potentially contain significant errors. Daily et al. [203] have formed a highly detailed model for NO LIF with over 950 discrete levels included. By forming such a detailed model, the influence of parameters such as RET, pressure, laser irradiance and final state EET under non-stationary conditions may be assessed in great detail. This data can potentially be used to validate reduced NO LIF models and assess the underlying assumptions in great detail. Rahmann et al. [204] and Kienle et al. [205] describe the LASKIN code for the detailed determination of linear regime LIF spectra from excited state OH. Significant attention has been paid in Brockhinke et al. [206, 207] to correctly model and validate the polarisation and spectral dependence of the OH LIF signal using LASKIN code. Generally, the detailed simulation of OH excitation dynamics using the rate equation approach has been utilised in several other studies [208, 209, 210, 211, 212], however simulation of OH excitation dynamics down to the rotational or spin split level for non laser coupled levels is typically not necessary when evaluating the trends of the spectrally integrated LIF signal. It is proposed in this section that a detailed six-level transient rate equation model is adequate for the purpose of validation of the spectrally integrated LIF signal considered in this study.

4.2.1 Model Overview

The detailed six-level model utilised as the reference case for the numerical studies in this section features discrete levels for the laser coupled rotational levels in both the ground state and the excited states. Two lumped levels represent the remainder of the laser coupled vibrational manifolds for J ≠ J′′ ′′, = and ν 0 J ≠J′ ′, = in the ground and ν 0 excited states respectively. A lumped level models the population transfer due to VET in the excited state from ν′ =1 to ν′ =0. Population storage in the ground-state non laser-coupled vibrational manifold is modelled by a lumped level representing the ν′′ =1 manifold. Schematically, the energy transfer diagram for the detailed six-level model showing the dominant energy transfer pathways and the numbering of the relevant levels is presented in Fig. 4-1.

Fig. 4-1. Energy transfer diagram for the six-level detailed model. All spontaneous emission paths from the excited A²Σ⁺ state as well as EET and VET pathways to and from levels 1 and 2 are not shown in this diagram for clarity.

As both the temporal and spatial variation in the irradiance is relevant, the population levels hence LIF signal will be a function of both space and time. Based on the spatial and temporal dependence of the irradiance the governing equations for the system are essentially a system of six partial differential equations (PDEs). However, this system of six PDEs is separable in space and time. After separation the system of PDEs can be reduced to a system of six ODEs for the temporal variation over a differential element of

the spatial co-ordinate in the laser sheet thickness direction (z) which is represented by Eqs. (4.1)-(4.6). The variation in the LIF signal through the laser sheet thickness is essentially not of interest, but rather the variation of the integrated signal for different irradiance distributions is. Only the integrated value of the LIF signal through the thickness of the laser sheet is considered by Eq.(4.8).

The use of normalized population level fraction Ni, is equivalent to the actual concentration in level i (ni), divided by the total OH concentration (Ni=ni/nOH). The initial conditions for Eqs(4.1)-(4.6) for all simulations is taken to be equilibrium governed by detailed balance. Equations (4.1)-(4.6) form a coupled system of ODEs and are solved using a high order explicit Runge-Kutta scheme with adaptive time stepping and error control. The absolute solution error is verified to be less than 5x10^-12 by substituting the computed values of Ni into Eq. (4.7) and examining the residual.

For all of the numerical comparisons made in this Chapter, the bath gas composition is taken to be the equilibrium combustion products of an atmospheric pressure methane-air flat flame at an equivalence ratio of 0.7 and a temperature of 1750K. The selection of this thermo-chemical state for the numerical validation is based on using a consistent bath gas composition with the experimental validation reported in Section 5. The Einstein Aji and Bij, and Bji coefficients are taken from LIFBASE [213]. The mild rotational level dependence of the Aij coefficients has been accounted for by calculating temperature dependant values for the lumped levels. The EET values for v'=0 are taken from Paul et al. [214] and the references contained within, additional values are taken from [215, 216, 217, 218]. The high temperature EET and VET values for ν′ =1 are taken from Paul [219], with additional data, particularly at lower temperatures taken from [220, 221].

Broadening of the absorption line width due to homogenous broadening was accounted for using the collisional cross sections reported by Rea et al. [222, 223] and Kessler et al.

[224].

For the detailed six-level model used in this study, an effective rate for RET is calculated such that the predicted RET rate is proportional to the degree of equilibrium departure.

As an example, the functional form for the RET term T61, in Eq. (4.1) is given in expanded form by Eq.(4.9), the T23 RET term in Eq. (4.2) can also be expressed in a similar fashion as Eq. (4.9), however with different subscripts for the relevant Ni terms.

An important feature of the RET model employed is that detailed balance requirements are satisfied. Although the RET model employed in this study is not as rigorous and universally applicable as RET schemes such as the energy corrected sudden scaling law proposed by Kienle et al. [225], for atmospheric pressures and nanosecond excitation combined with the use of a lumped six-level model, the RET model in this study is deemed to be acceptably accurate. Based on the limited data available for RET, the results and values proposed in several studies [204, 212, 216, 226, 227] are combined and used to calculate the RET cross section. For the bath gas composition investigated, a value for the RET cross section of approximately 50Ǻ is predicted.

The spectrally, temporally and spatially (along the z-axis) integrated fluorescence signal (F) is of primary interest in this study and is defined by Eq.(4.8). In Eq. (4.8) double summation over all the upper states (i=2,3,4) and lower states (j=1,5,6) is implied. The use of Tij notation in Eqs. (4.1)-(4.6) corresponds to a simplified notation, with each Tij

term representing the combined values of RET, EET, VET and spontaneous emission where relevant.

4 the laser, φL(ν), and absorption φa(ν) line-width functions are normalised to be consistent with the normalisation scheme of Partridge and Laurendeau [228]. The spectral irradiance

(

^{, ,}

)

I_ν ν z t is decomposed into the normalised spectral irradiance (I_ν⁰) and the relevant normalised spectral, spatial and temporal variation functions as shown by Eq.(4.10).

(

^{, ,}

)

⁰ z

( ) ( )

I_ν ν z t =I f z f t_ν (4.10)

As Q-branch transitions are examined experimentally in this study, the degeneracies of the excited and ground state excited rotational levels are equal for the detailed six-level model. An example of the terms involved in the pumping rate term (W12) is given by Eq.(4.11). The φ_ν′′ term in Eq. (4.11) represents the dimensionless integral overlap term

for the ground state, in the idealised monochromatic laser and absorption limit this term will be unity, however for the real broadened system this term will be less than unity but greater than zero.

To model the temporal variation of the laser pulse f tt

( )

, a quadratic exponential function is selected. The quadratic exponential functional form given by Eq. (4.12) is found to be a good functional representation for the temporal variation of a frequency doubled dye laser pumped by a frequency doubled Nd:YAG laser for all time periods during the laser pulse. The value of ∆t in Eq. (4.12) is set to a value of ∆ = ∆t t_FWHM C_QE for the simulations, by doing so the FWHM and the integral for t>0 of Eq. (4.12) is equal to

tFWHM

∆ . An explicit analytic expression for the normalising constant C is not possible _QE to be derived, therefore C is calculated numerically and a value correct to five _QE significant figures is found to be 1.1549. The value of ∆tFWHM is set to 8ns, which is the same FWHM as the laser pulse used for the experimental validation in Section 4.4.

( ) (

⁴

) ⁽ ⁾

²^exp

⁽ ⁾

t QE

f t =C π t t∆ ^⎡⎣− ∆t t ^⎤⎦ (4.12)

4.2.2 Assessment of RET and Final State Effects

For the detailed six level model there is a significant degree of uncertainty in value of the RET cross section, this is due to the large variation in the reported RET cross sections found in literature [204, 206, 207, 212, 216, 226, 227, 229]. As illustrated by Tobai et al.

[226], the disparities between data sets for RET cross sections are often so large that the agreement between studies is often not possible within the quoted error or uncertainty bars. To assess the potential influence of the uncertainty in the RET rate (R) on the

detailed six-level model results, the relative sensitivity of the spectrally and temporally integrated fluorescence signal F, to the RET rate is computed, this is abbreviated as S_{F R}^rel_, . The value of S_{F R}^rel_, is computed using the definition S_{F R}^rel_, = ∂lnF ∂lnR, which is effectively the linearised relative sensitivity of F to R, evaluated at the selected RET rate.

To evaluate S_{F R}^rel_, for different irradiance regimes S_{F R}^rel_, is computed as a function of the normalised spectral irradiance, a plot of S_{F R}^rel_, vs. I_ν⁰ is shown in Fig. 4-2.

Fig. 4-2. Relative sensitivities of the temporally integrated LIF signal, F, as a function of the normalized spectral irradiance (I_ν⁰). Dashed line shows the relative sensitivity of F to the rotational energy transfer rate (S_{F R}^rel_, ). Solid line shows the relative sensitivity of F to the quenching loss factor (S_{F L}^rel_, ).

From Fig. 4-2, at low irradiance levels where I_ν⁰<1x10^-6, S_{F R}^rel_, is <1x10^-3, hence the sensitivity of F to the RET rate is small and is not a significant factor in this irradiance regime. This low sensitivity at low irradiance levels is caused by there being only a small fraction of the laser coupled level participating in the LIF process, hence replenishment of population in the laser coupled level by RET is not important. For increasingly large laser irradiances, an increased proportion of the laser coupled level participates in the LIF process, RET becomes increasingly important to replenish the population in the laser

coupled level from the other rotational levels ground state ν′′ =0 rovibrational manifold.

Hence the normalised sensitivity is larger and increases with increasing irradiance as indicated by the region of constant positive slope for I_ν⁰ <1x10⁷ W/(cm².cm^-1) in Fig.

4-2. At very high irradiance levels, Fig. 4-2 shows that the sensitivity of the fluorescence signal to RET remains relatively high, but plateaus around 1x10⁸ W/(cm².cm^-1) rather than continuing to increase with increasing irradiance levels. At these high irradiance levels the limiting factor is no longer the rate at which RET can replenish the laser coupled level from the ground state ν′′ =0 rovibrational manifold, but rather the rate at which population is transferred into the ground state ν′′ =0 manifold. At high irradiance levels depletion of the entire ground state ν′′ =0 rovibrational manifold is the limiting factor minimising any further increase in S_{F R}^rel_, with increasingly large values of I_ν⁰.

The largest difference in the RET rate reported in literature compared to the value used in this study is approximately 50%. Considering a PLIF experiment that utilises very high normalised spectral irradiance levels such as 1x10⁹ W/(cm².cm^-1), based on the sensitivity results reported in Fig. 4-2, an error of only 2% in the absolute fluorescence signal is predicted. This error would only be realised if an absolute calibration method were utilised such as Rayleigh scattering [190]. If a relative calibration were utilised such as a known OH concentration in a flat flame, a much smaller error due to the RET uncertainty would result. With such a small difference in the absolute LIF signal for a 50% change in the RET rate, the uncertainty in the value of the RET rate derived from literature is not significant for the spectrally integrated results presented in this study.

A possible source of error in the detailed six-level model simulations of the OH LIF process is the uncertainty in the final state after electronic quenching. There are three possible paths that are different to the path presented in Fig. 4-1 that an excited A²Σ⁺ state molecule can possibly end up at after an electronic quenching event. The implications and assumptions made by neglecting these three paths are discussed below.

After an electronic quenching event the OH molecule can end up in a chemically modified or ionised state, the recovery time from such a state to chemical equilibrium levels is typically of the order of microseconds or more at atmospheric pressure, therefore the molecule is effectively lost from the LIF population cycle during the time scale of the laser pulse. In order to assess the sensitivity of the solution to population loss due to final state quenching, an additional level N7, has been added to the system of equations for the detailed six-level model to account for the lost population due to electronic quenching, this equation is given by Eq. (4.13).

( )

7 25 2 35 3 46 _B 0, 41 4

dN LQ N LQ N L Q f v J J Q N

dt = + + ⎡⎣ + ′= = ′′ ⎤⎦ (4.13)

All quenching values contributing to the Tij terms in Eqs. (4.1)-(4.6) have been changed from the original Qij form to (1-L)Qij, thus accounting for the quenching population loss factor L. The value of the quenching loss factor L, is taken to be a global factor modelling the rate of quenching loss, the value of L varies from zero for no population loss to unity for all population lost after electronic quenching. The linearised sensitivity of F to the quenching loss factor L (S_{F L}^rel_, ), is presented as a function of the normalised spectral irradiance in Fig. 4-2. An approximately similar trend for the loss factor sensitivity S_{F L}^rel_, as a function of irradiance is found for RET rate sensitivityS_{F R}^rel_, , both of these sensitivities are shown on the same graph in Fig. 4-2. For irradiance levels below 1x10⁷ W/(cm².cm^-1) Fig. 4-2 shows that there is a steady increase in the sensitivity S_{F L}^rel_, with increasing irradiance due to the increasing significance of population cycling effects.

Although the rate of increase of the sensitivity S_{F L}^rel_, does reduce at high irradiance levels, there is not the same degree of levelling out as S_{F R}^rel_, at high irradiance levels. The continued increase in sensitivity of S_{F L}^rel_, at high irradiance levels is proposed to be due to the large degree of population cycling that occurs at high irradiance levels.

By considering a loss factor of 0.1 and comparing the absolute LIF signal obtained at a normalised spectral irradiance level of 1x10⁹ W/(cm².cm^-1), a 0.1 loss factor will reduce the LIF signal by 12% compared to a loss factor of zero. This is a significant variation and if accurate non-linear LIF measurements are required to be made using absolute calibration methods such as Rayleigh scattering [190], then accurate temperature and species dependant values for the quenching loss factor will be required. Relative calibrations will reduced the uncertainty introduced by the quenching loss factor;

however these uncertainties will not be completely mitigated.

Although there have been no published experimental measurements of the quenching loss factor for OH at flame temperatures that the authors are aware of, there has been measurements of final state OH at room temperature by Crosley and Copeland [230] and branching ratios of NO at room temperature by Settersten et al. [231]. These studies of small diatomic molecules indicate that some population is lost for room temperature quenching, however there is no theory that describes this process well and extrapolations of the experimental results to flame temperatures is not well justified, particularly as it is known the dominant quenching mechanism is fundamentally different at low and high temperatures for OH. For the detailed six-level model in the remainder of this Chapter a loss factor of zero is assumed. No specific value for the loss factor for OH at flame temperature has been reported experimentally or theoretically. The introduction of a loss factor greater than zero makes it virtually impossible to form detailed reduced analytic models.

A Second possible outcome after an electronic quenching event is that the molecule may end up in a high vibrational level in the ground state such as ν′′ ≥2. If this is the case energy transfer processes maybe too slow to return this molecule to the laser coupled level within the time scale of the laser pulse. Measurements at room temperature of the VET rate for high vibrational levels in the ground state of OH by Dyer et al. [232] has shown that this process can be quite slow relative to quenching and excited state VET.

No published study that these authors are aware of has explicitly examined VET cross sections or decay rates in the ground state of OH for any vibrational level at flame

temperatures. In a study using a two-colour LIF pump-probe experimental setup to examine ground state RET in a stoichiometric methane-air flame, Chen et al. [227]

indirectly show that VET from ν′′ =1 is slow relative to RET. Absolute determination of the ground state VET cross sections or decay rate from ν′′ =1 cannot be determined from the results presented by Chen et al. [227], however a lifetime longer than 3ns can be inferred.

The significance of population in high vibrational levels in the ground state due to final state quenching has been modelled by adding additional levels to the detailed six-level model, each of the additional levels represent a vibrational level in the ground state up to ν′′ =10. Based on the indirect results of Chen et al. [48] we assume that the VET rate for ν′′ ≥2 will be less than or equal to the VET rate from ν′ =1 at flame temperatures, thus VET cross sections were assumed to be identical to those from ν′ =1 in the absence of any other available data. The high vibrational levels in the ground state are populated by controlling the branching ratio for the final state of quenching; this effectively controls the ratio of the population that arrives in high vibrational levels to the population that arrives in levels where the vibrational quantum number is conserved. A parameter is also used to control the value of the high vibrational level the population is assumed to be

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